not very hard and not easy, nice geometry

An angle is given (<180 degree) with a vertex X and a point P placed inside this angle. Points A and B lie on different arms of this angle, and XA=XB and the sum PA+PB is the smallest. I ask: when this sum is the smallest? Maybe the angles APX and BPX are equal??

I assume that are fixed points such that and is a variable point such that is smallest.
Then and in this case
If , then . can be anywhere on the segment , so is not necessary that
Also, the condition is not necessary.

An angle is given (<180 degree) with a vertex X and a point P placed inside this angle. Points A and B lie on different arms of this angle, and XA=XB and the sum PA+PB is the smallest. I ask: when this sum is the smallest? Maybe the angles APX and BPX are equal??

It is not so easy, that's why i placed it here. Can anybody help me?

If (PA + PB) is the smallest, then (PA + PB) must be a straight line. The shortest distance between two points, A and B here, is a straight line.

That means P is anywhere along the line AB.

That means then that angles APX and BPX are supplementary----their sum is 180 degrees.